Finite Group Databasebeta

← #241366

Subgroups — #241366

Order 512.

906 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35239C211yesmaximal, characteristic
2#21987C211yesmaximal, characteristic
3C2×#1948C211yesmaximal, characteristic
4#20800C211yesmaximal, characteristic
5#10580C212yesmaximal, complemented
6#10579C212yesmaximal, complemented
7#10576C212yesmaximal, complemented
8#10574C212yesmaximal, complemented
9#8044C211yesmaximal, characteristic
10#7904C211yesmaximal, characteristic
11#7895C211yesmaximal, characteristic
12C2×#432C2211yescharacteristic
13C2×#428—22no—
14C22×#106—21no—
15C22×C4⋊C8C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2857—21no—
18#2851—21no—
19#2666—22no—
20#2571C2211yescharacteristic
21#2570—21no—
22#2552C2211yescharacteristic
23#2552C2211yescharacteristic
24#2551C2211yescharacteristic
25#2550—21no—
26#2442—21no—
27#2440—21no—
28#2438—22no—
29#2331—21no—
30#2330—21no—
31#2285C2211yescharacteristic
32C2×#384C2211yescharacteristic
33#2283C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1948—21no—
36#1948—21no—
37#1948C2212yes—
38#1948C2212yes—
39#1948C2212yes—
40#1947C2212yes—
41#1853—21no—
42#1852—21no—
43#1847—22no—
44#1847—21no—
45#1845C2212yes—
46#1844C2212yes—
47#1836—21no—
48C2×#340—21no—
49#1296C2212yes—
50#1296C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55C22×M16—21no—
56C23×C8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×#125—21no—
60C2×#125—21no—
61C2×C22⋊Q8—22no—
62C2×#117D412yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#432—22no—
71#432Q814yescomplemented
72#428—48no—
73C2×#107—24no—
74C2×#107—22no—
75C2×#106—24no—
76C2×#106—22no—
77C2×#106—22no—
78C2×#106—22no—
79C2×#106—44no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8Q814yescomplemented
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×C22⋊C8—21no—
86C4×M16—22no—
87C4×M16Q814yescomplemented
88C2×#97—21no—
89C2×#97—21no—
90C2×C4×C8C2312yes—
91#400—22no—
92#400—22no—
93#398—22no—
94#398—22no—
95#398—22no—
96#398—22no—
97#395—22no—
98#395—22no—
99#391—22no—
100#391—22no—
101#389—22no—
102#389—22no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#385—22no—
116#385—22no—
117#384—21no—
118#384—21no—
119#384—21no—
120#384—21no—
121#384—21no—
122#384C2312yes—
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#383—21no—
126#382—21no—
127#382—21no—
128#382D411yescharacteristic
129#382D411yescharacteristic
130#382D412yes—
131#381—21no—
132#380—22no—
133#380—21no—
134#380—21no—
135#380D411yescharacteristic
136#380D411yescharacteristic
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4D411yescharacteristic
142C4×C4⋊C4D411yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145C4×C4⋊C4C2311yescharacteristic
146C4×C4⋊C4C2311yescharacteristic
147C4×C22⋊C4—21no—
148#376—21no—
149C2×#95—21no—
150#340—22no—
151#340—22no—
152#340—22no—
153#340—22no—
154C22×Q8—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—41no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C2×M16—24no—
162C2×M16—24no—
163C2×M16—24no—
164C22×C8—22no—
165C22×C8—24no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—42no—
169C22×C8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—82no—
173#125—42no—
174#125—42no—
175#125—82no—
176C22⋊Q8—84no—
177#117—44no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—82no—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×Q812yes—
232C2×C42C2×Q812yes—
233C2×C42C2×Q812yes—
234C2×C42#4112yes—
235C2×C42C2411yescharacteristic
236#107—48no—
237#107—82no—
238#106—44no—
239#106—48no—
240#106—44no—
241#106—82no—
242C4⋊C8—24no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8C2×Q814yes—
246C4⋊C8C2×Q814yes—
247C22⋊C8—24no—
248#97—24no—
249#97—24no—
250C4×C8—22no—
251C4×C8C2×Q814yes—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—22no—
267#95—22no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272C24#12011yescharacteristic
273C2×Q8—44no—
274C2×Q8—82no—
275C2×Q8—82no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—81no—
300C22×C4—81no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4#12011yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4C22⋊Q812yes—
308C22×C4C22⋊Q812yes—
309C22×C4C22⋊Q812yes—
310C22×C4C4⋊Q812yes—
311C22×C4C4⋊Q812yes—
312C22×C4C4⋊Q812yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316M16—216no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—48no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4AGL(1,ℤ/8)12yes—
365C4⋊C4#13712yes—
366C22⋊C4—44no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—42no—
374C42—42no—
375C42—81no—
376C42—81no—
377C42—82no—
378C42C22⋊Q812yes—
379C42C22⋊Q812yes—
380C42C22⋊Q812yes—
381C42C4⋊Q812yes—
382C42C4⋊Q812yes—
383C42C4⋊Q812yes—
384C42C22×Q811yescharacteristic
385C42C22×Q811yescharacteristic
386C42C22×Q811yescharacteristic
387C42C2×#4111yescharacteristic
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23#39512yes—
393C23#45312yes—
394C23#45612yes—
395C23C2×#12011yescharacteristic
396Q8—164no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—164no—
449C2×C4#39512yes—
450C2×C4#39512yes—
451C2×C4#39512yes—
452C2×C4C2×C22⋊Q811yescharacteristic
453C2×C4C2×C22⋊Q811yescharacteristic
454C2×C4C2×C22⋊Q811yescharacteristic
455C2×C4C2×C4⋊Q811yescharacteristic
456C2×C4C2×C4⋊Q811yescharacteristic
457C2×C4C2×C4⋊Q811yescharacteristic
458C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
459C2×C4C2×#13711yescharacteristic
460C2×C4#57511yescharacteristic
461C2×C4#57511yescharacteristic
462C2×C4#57712yes—
463C2×C4#57712yes—
464C8—28no—
465C8—44no—
466C8—44no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22#206014yes—
474C22C2×#39511yescharacteristic
475C22#300611yescharacteristic
476C22#300711yescharacteristic
477C22#300911yescharacteristic
478C22#301011yescharacteristic
479C22C2×#45311yescharacteristic
480C22C2×#45611yescharacteristic
481C4—22no—
482C4—22no—
483C4—21no—
484C4—21no—
485C4—22no—
486C4—42no—
487C4—42no—
488C4—81no—
489C4—82no—
490C4—81no—
491C4—81no—
492C4—82no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—162no—
497C4#307711yescharacteristic
498C4#307811yescharacteristic
499C4#308412yes—
500C4#316312yes—
501C4#316511yescharacteristic
502C4#316611yescharacteristic
503C2—24no—
504C2#1856311yescharacteristic
505C2#1856411yescharacteristic
506C2#1858611yescharacteristic
507C2#1858911yescharacteristic
508C2#2355411yescharacteristic
509C2#2355611yescharacteristic
510C2C2×#206011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #10580C2semidirect
6. #10579C2semidirect
7. #10576C2semidirect
8. #10574C2semidirect
71. #432Q8semidirect
82. C2×C4⋊C8Q8semidirect
87. C4×M16Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#352392
#219872
C2×#19482
#208002
#105802
#105792
#105762
#105742
#80442
#79042
#78952